|ImageDescElem=Polar equations are used to create interesting curves, and in most cases they are periodic like sine waves. Other types of curves can also be created using polar equations besides roses, such as Archimedean spirals and limaçons. See the [[Polar Coordinates]] page for some background information.

|ImageDescElem=Polar equations are used to create interesting curves, and in most cases they are periodic like sine waves. Other types of curves can also be created using polar equations besides roses, such as Archimedean spirals and limaçons. See the [[Polar Coordinates]] page for some background information.

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|ImageDesc=== Rose ==

|ImageDesc=== Rose ==

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The general polar equations form to create a rose is <math>r = a \sin(n \theta)</math> or <math>r = a \cos(n \theta)</math>. Note that the difference between sine and cosine is <math>\sin(\theta) = \cos(\theta-\frac{\pi}{2})</math>, so choosing between sine and cosine affects where the curve starts and ends. <math>a</math> represents the maxium value <math>r</math> can be, i.e. the maximum radius of the rose. <math>n</math> affects the number of petals on the graph:

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The general polar equations form to create a rose is <math>r = a \sin(n \theta)</math> or <math>r = a \cos(n \theta)</math>. Note that the difference between sine and cosine is <math>\sin(\theta) = \cos(\theta-\frac{\pi}{2})</math>, so choosing between sine and cosine affects where the curve starts and ends. <math>a</math> represents the maximum value <math>r</math> can be, i.e. the maximum radius of the rose. <math>n</math> affects the number of petals on the graph:

<p>

<p>

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* If <math>n</math> is an odd integer, then there would be <math>n</math> petals, and the curve repeats itself every <math>\pi</math>.

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* If <math>n</math> is an odd integer, then there would be <math>n</math> petals, and the curve repeats itself every <math>\pi</math>.<br>'''Examples: '''{{hide|1=

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* If <math>n</math> is an even integer, then there would be <math>2n</math> petals, and the curve repeats itself every <math>2 \pi</math>.

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:: [[Image:Odd2.jpeg|600px]]}}

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* If <math>n</math> is a rational fraction (<math>p/q</math> where <math>p</math> and <math>q</math> are integers), then the curve repeats at the <math>\theta = \pi q k </math>, where <math>k = 1</math> if <math>pq</math> is odd, and <math>k = 2</math> if <math>pq</math> is even.

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* If <math>n</math> is an even integer, then there would be <math>2n</math> petals, and the curve repeats itself every <math>2 \pi</math>.<br>'''Examples: '''{{hide|1=

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* If <math>n</math> is irrational, then there are an infinite number of petals.

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:: [[Image:Even.jpeg|600px]]}}

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<br>'''Below is an applet to graph polar roses:'''<br>{{Hide|1=

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* If <math>n</math> is a rational fraction (<math>p/q</math> where <math>p</math> and <math>q</math> are integers), then the curve repeats at the <math>\theta = \pi q k </math>, where <math>k = 1</math> if <math>pq</math> is odd, and <math>k = 2</math> if <math>pq</math> is even.<br>'''Examples: '''{{hide|1=

{{!}}Archimedes' Spiral<br> <math> r = a\theta </math> <br> [[Image:Archimedes' spiral.png|350px]]<br>The spiral can be used to square a circle, which is constructing<br> a square with the same area as a given circle, and trisect an<br> angle, which is constructing an angle that is one-third of a given <br>angle (more on these topics can be found under [http://mathforum.org/mathimages/index.php/Polar_Equations#Related_Links related links] ).{{!}}{{!}}Fermat's Spiral<br> <math> r = \pm a\sqrt\theta </math> <br> [[Image:fermat's_spiral.jpg|350px]]<br> This spiral's pattern can be seen in disc phyllotaxis,<br> which is the circular head in the middle of flowers <br>(e.g. [http://mathforum.org/mathimages/index.php/Polar_Equations#Why_It.27s_Interesting sunflowers]).

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{{!}}Hyperbolic spiral<br><math> r = \frac{a}{\theta}</math><br> [[Image:Hyperbolic_spiral.png|350px]] <br>It begins at an infinite distance from the pole, and <br>winds faster as it approaches closer to the pole.{{!}}{{!}}Lituus<br> <math> r^2 \theta = a^2 </math><br>[[Image:Lituus.png|400px]]<br>It is asymptotic at the <math>x</math> axis as the distance increases <br>from the pole.

{{!}}Hyperbolic spiral<br><math> r = \frac{a}{\theta}</math><br> [[Image:Hyperbolic_spiral.png|350px]] <br>It begins at an infinite distance from the pole, and <br>winds faster as it approaches closer to the pole.{{!}}{{!}}Lituus<br> <math> r^2 \theta = a^2 </math><br>[[Image:Lituus.png|400px]]<br>It is asymptotic at the <math>x</math> axis as the distance increases <br>from the pole.

To find the '''area''' of a sector of a circle, where <math> r </math> is the radius, you would use <math> A = \frac{1}{2} r^2 \theta </math>. <br>

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[[Image:CircleArea.png|Area of a sector of a circle.|thumb|200px|left]]To find the '''area''' of a sector of a circle, where <math> r </math> is the radius, you would use <math> A = \frac{1}{2} r^2 \theta </math>. [[Image:PolarArea.png|<math>A = \int_{-\frac{\pi}{4}}^\frac{\pi}{4}\! \frac{1}{2} \cos^2(2\theta) d\theta</math>|thumb|200px|right]]<br>

Therefore, for <math> r = f(\theta)</math>, the formula for the area of a polar region is: <br>

Therefore, for <math> r = f(\theta)</math>, the formula for the area of a polar region is: <br>

Polar coordinates are often used in navigation, such as aircrafts. They are also used to plot gravitational fields and point sources. Furthermore, polar patterns are seen in the directionality of microphones, which is the direction at which the microphone picks up sound. A well-known pattern is the [[Cardioid]].

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|Field=Algebra

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</p>Archimedes' spiral can be used for compass and straightedge division of an angle into <math>n</math> parts and circle squaring. <ref name=spiral> Weisstein, Eric W. (2011). http://mathworld.wolfram.com/ArchimedesSpiral.html. Wolfram:MathWorld.</ref> Fermat's spiral is a Archimedean spiral that is observed in nature. The pattern happens to appear in the mesh of mature disc phyllotaxis. The florets in sunflowers are arranged in a form of that spiral. Archimedean spirals can also be seen in patterns of solar wind and Catherine's wheel.

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|Field2=Calculus

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<p>As you can see, these equations can create interesting curves and patterns. More complicated patterns can be created with more complicated equations, like the image on the right. Since intriguing patterns can be expressed mathematically, like these curves, they are often used for art and design.

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|WhyInteresting=Polar coordinates are often used in navigation, such as aircrafts. They are also used to plot gravitational fields and point sources. Furthermore, polar patterns are seen in the directionality of microphones, which is the direction at which the microphone picks up sound. A well-known pattern is a cardioid.<br><br>

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==Possible Future work==

==Possible Future work==

* More details can be written about the different curves, maybe they can get their own pages.

* More details can be written about the different curves, maybe they can get their own pages.

* Applets can be made to draw these different curves, like the one on the page for roses.

* Applets can be made to draw these different curves, like the one on the page for roses.

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|FieldLinks=[[Polar Coordinates]]<br>[[Cardioid]]

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* A page should be made to expand on squaring a circle and another on trisecting an angle, since they are both ancient math problems.

Basic Description

Polar equations are used to create interesting curves, and in most cases they are periodic like sine waves. Other types of curves can also be created using polar equations besides roses, such as Archimedean spirals and limaçons. See the Polar Coordinates page for some background information.

A More Mathematical Explanation

Rose

The general polar equations form to create a rose is UNIQ1d550f31571b57d9-math-00000001-Q [...]

[Click to hide A More Mathematical Explanation]

Rose

The general polar equations form to create a rose is or . Note that the difference between sine and cosine is , so choosing between sine and cosine affects where the curve starts and ends. represents the maximum value can be, i.e. the maximum radius of the rose. affects the number of petals on the graph:

If is an odd integer, then there would be petals, and the curve repeats itself every .Examples: [show more][hide]

If is an even integer, then there would be petals, and the curve repeats itself every .Examples: [show more][hide]

If is a rational fraction ( where and are integers), then the curve repeats at the , where if is odd, and if is even.Examples: [show more][hide]

The angle coefficient is . , which is even. Therefore, the curve repeats itself every

The angle coefficient is . , which is odd. Therefore, the curve repeats itself every

If is irrational, then there are an infinite number of petals.Examples: [show more][hide]

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Below is an applet to graph polar roses, which is used to graph the examples above:[show more][hide]

Other Polar Curves

Archimedean Spirals

Archimedes' SpiralThe spiral can be used to square a circle, which is constructing a square with the same area as a given circle, and trisect an angle, which is constructing an angle that is one-third of a given angle (more on these topics can be found under related links ).

Fermat's Spiral This spiral's pattern can be seen in disc phyllotaxis, which is the circular head in the middle of flowers (e.g. sunflowers).

Hyperbolic spiralIt begins at an infinite distance from the pole, and winds faster as it approaches closer to the pole.

LituusIt is asymptotic at the axis as the distance increases from the pole.

Limaçon[1]
The word "limaçon" derives from the Latin word "limax," meaning snail. The general equation for a limaçon is .

Why It's Interesting

Polar coordinates are often used in navigation, such as aircrafts. They are also used to plot gravitational fields and point sources. Furthermore, polar patterns are seen in the directionality of microphones, which is the direction at which the microphone picks up sound. A well-known pattern is the Cardioid.

Archimedes' spiral can be used for compass and straightedge division of an angle into parts and circle squaring. [4] Fermat's spiral is a Archimedean spiral that is observed in nature. The pattern happens to appear in the mesh of mature disc phyllotaxis. The florets in sunflowers are arranged in a form of that spiral. Archimedean spirals can also be seen in patterns of solar wind and Catherine's wheel.

As you can see, these equations can create interesting curves and patterns. More complicated patterns can be created with more complicated equations, like the image on the right. Since intriguing patterns can be expressed mathematically, like these curves, they are often used for art and design.

Possible Future work

More details can be written about the different curves, maybe they can get their own pages.

Applets can be made to draw these different curves, like the one on the page for roses.

A page should be made to expand on squaring a circle and another on trisecting an angle, since they are both ancient math problems.